Interesting pdf question

[ladyluck]

Z1,....,Z12 are iid standard normal

What is the pdf of

Z1/(Z2+Z3) + Z4/(Z5+Z6) + Z7/(Z8+Z9) + Z10/(Z11+Z12)

My thoughts are the denominators are N~(0,2). The numerators are N~(0,1). The ratio is probably cauchy. Any ideas on elegant ways to solve?

[Dason]

If you play around with the multiplying constants you could get each of the 4 terms to have the form K*X where X is a standard Cauchy. Then you could probably use the fact that the average of a bunch of standard Cauchy variables just gives you a standard cauchy (that is if you've shown this already...) to simplify a little further.

[ltleung]

this article should help, D. V. Hinkley (December 1969). "On the Ratio of Two Correlated Normal Random Variables". Biometrika 56 (3): 635–639.
Substitute the corresponding parameters.